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Transcript
```An Aerodynamicist’s
View of Lift, Bernoulli,
and Newton
Charles N. Eastlake
T
ake out a pencil, please. Here is a short
quiz. The sole question is multiple choice.
The production of lift by an airfoil is described correctly and accurately by:
A. Bernoulli’s Law
B. Newton’s Law(s)
D. All of the above.
I believe that the answer is D. A and B are certainly
true. You will have to judge C.
As a career aerodynamicist, I am pleased to see flying machines discussed in physics classes. Their power
in captivating the imagination and motivating students is significant. As an occasional reader of and
contributor to TPT,1 however, I am dismayed to read
articles delving into the details of why Bernoulli’s law
is at fault in explaining how airfoils generate lift and
why Newton’s laws need to be used instead. Aerodynamics as taught to aeronautical engineers does not
find any controversy in this issue. Both approaches are
right in all situations. Which approach should be employed is merely a matter of which is more convenient
to use given the type of data available to characterize a
particular flow pattern. The inability of one model or
the other to explain lift production is really a problem
of using a version of these laws that is oversimplified.
In other words, a message worth emphasizing in
physics classes is that too many simplifying assumptions might ruin the accuracy of a theoretical model
that is fundamentally a correct choice and really
should work just fine.
Before we can look precisely at lift production, we
need to delve into some definitions of basic concepts
and terminology. Please note right up front that this is
an aerodynamicist’s view. Other technical fields that
are heavily involved with fluid flow sometimes have
terminology traditions that are a little different, partic-
Charles Eastlake has been an aeronautical engineer and pilot for 35 years. His industry
experience includes supersonic military aircraft, general aviation aircraft, jet engines, missiles, spacecraft, and race cars. He has taught aerodynamics and aircraft design in the
Aerospace Engineering Department at Embry-Riddle Aeronautical University for 23 years.
He wrote the textbook for the ERAU wind-tunnel-testing course and is the technical editor
of the current edition of the SAE Dictionary of Aerospace Engineering. He is an award-winning teacher and is active in science outreach programs. For the past three seasons he has
flown his Citabria in International Aerobatic Club competition.
Charles N. Eastlake
166
Department of Aerospace Engineering, Embry-Riddle Aeronautical University,
600 S. Clyde Morris Blvd., Daytona Beach, FL 32114-3900; [email protected]
THE PHYSICS TEACHER ◆ Vol. 40, March 2002
Fig. 1. Airfoil terminology.
ularly those focusing on pumping liquids through
pipes.
Airfoil Terminology
Referring to Fig. 1, let’s start with airfoil terminology. The farthest forward point on the airfoil shape is
called the leading edge, while the farthest aft point is
called the trailing edge. A straight line from leading
edge to trailing edge is called the chord. The curved
line defining the top half of the airfoil is called the upper surface, and the bottom half is called the lower surface. A curved line midway between upper and lower
surface is called the mean line, referring to arithmetic
mean of the upper and lower surface vertical-position
(measured perpendicular to the chord) coordinates.
Some texts call it the mean camber line, but mean line
seems like a more precise term to me. The greatest vertical distance between the mean line and the chord line
is called camber and is usually a few percent of the
length of the chord. The nose-up or nose-down attitude of the airfoil is described by angle of attack,
which is the angle between the chord line and the direction of the air movement relative to the airfoil (relative velocity vector).
The definition of lift and drag forces includes a
fine point relating to force direction that might slip
unnoticed through verbal discussions but becomes
critical when solving equations. There is a single resultant force acting on an airfoil. However, it can be resolved into components with respect to any convenient coordinate system. Aerodynamicists are usually
interested in what are known as wind axes, meaning
axes oriented with the relative velocity vector, also
known as relative wind. Lift is the force component
perpendicular to the relative wind, and drag is the
force component parallel to the relative wind. Lift and
drag vectors are often sketched as perpendicular and
THE PHYSICS TEACHER ◆ Vol. 40, March 2002
167
parallel to the chord of the airfoil, which is generally
not quite right. And lift and drag are even more frequently sketched as perpendicular and parallel to the
ground, which is correct only if one presumes that the
wing is flying level and thus is not correct if the
wing/airplane is climbing or descending.
Fluid Flow
With terminology squared away, we move on to
the basic physical principles governing fluid flow.
Analysis of fluid flow is typically presented to engineering students in terms of three fundamental principles: conservation of mass, conservation of momentum, and conservation of energy. Newton’s name seldom comes up, except to encourage student ownership of these principles by reminding the class that certain segments of them, particularly conservation of
linear momentum, are what they probably called Newton’s laws in physics class.
The equations that represent these principles can
be expressed in two ways. High school students are
probably not prepared to absorb this point, but college
students might find it interesting. The three laws can
be expressed as a set of simultaneous differential equations or integral equations. The differential approach
is most useful if one intends to describe the fluid behavior at a specific location, or at many locations in
the process of mapping out the details of the flow field.
This is where Bernoulli’s law is most frequently
brought into the picture. This is also the foundation
of most computer programs such as FoilSim.2 The integral approach, which sometimes becomes recognizable as Newton’s laws, concentrates on the larger-scale
phenomena of what changes in momentum and energy occurred in a region of the flow, and what forces
had to be exerted on the flow to make those changes
occur. This model is dramatically simpler for certain
types of problems like thrust produced by a jet engine
because the complex details of the behavior of the flow
inside the engine do not appear in the equations. Both
approaches are equally valid and equally correct, a concept that is central to the conclusion of this article.
Conservation of Energy
First consider conservation of energy, since this is
where Bernoulli’s law shows up. (Here’s a possible injection point for interdisciplinary/historical content.
168
Have your students research Bernoulli’s background.
Bernoulli is actually three guys, a real family affair.) A
streamline is the path a particle follows as it moves
through a flow field. Visualize a time-lapse photo of
car headlights moving along a highway at night. The
light streaks are streamlines. The total energy of a fluid “particle” or infinitesimally small volume of fluid is
constant as it travels along a streamline if no external
work is done on it. This is generally true for external
flows like air flowing around an airfoil. In contrast,
for internal flows like flow pumped through pipes or
flow through the inside of that jet engine, the total energy might not be constant. In either case energy can
be shifted back and forth between kinetic energy and
potential energy as the particle moves.
The measurable characteristic that quantifies total
energy is called total pressure, PT , which is measured
with a Pitot tube, also known simply as a total pressure
tube. This type of pressure instrument is nothing
more than a tube with the open end facing directly upstream so that it “catches” the moving air. (Another
interesting spot for some interdisciplinary/historical
content. Have students look up Henri Pitot and see
what he was doing when he devised this instrument.
He was measuring water flow around bridge pilings in
the Seine River in Paris, and proved that the accepted
flow theory of the time was seriously in error.) If you
do external work on a fluid, like putting it through a
pump, you raise its total pressure. If you make the fluid do external work, like having it turn a windmill,
you reduce its total pressure.
Potential energy is quantified by static pressure, PS.
Static pressure is the pressure component that is not influenced by the direction of movement. Some references say that if you simply say “pressure” that “static
pressure” is clearly implied. I have found that this
shorthand causes confusion and thus I never use it. In
concept it is measured by drilling a measurement hole
in a surface that is parallel to the velocity vector. In a
wind tunnel, this is trivially easy. One simply drills a
hole in the wall of the tunnel. On an airplane in flight,
it can be tricky because of interference of the flow
around several nearby parts of the airplane, but a hole
in the side of the fuselage is still the usual solution.
Another alternate terminology issue may arise here.
As previously mentioned, a pump raises the total pressure of the fluid. In flows contained by walls such as
THE PHYSICS TEACHER ◆ Vol. 40, March 2002
pipes, the energy addition also raises the static pressure. And static pressure in a pipe is easier to measure
than total pressure, so there is a tradition of describing
work done on the fluid as an increase in static pressure.
In external flows like flow over airfoils, the energy addition process is fundamentally different. Work done
on this flow usually appears in the form of an increase
in velocity, like you feel when standing in front of a
view.
Kinetic energy is quantified by dynamic pressure,
which is the product of fluid density and velocity
squared, ½ ␳V 2. Comparing it to ½mv 2, dynamic
pressure is clearly the familiar definition of kinetic energy divided by fluid volume. Since it occurs over and
over in aerodynamic discussions, it is usually abbreviated as q.
In low-speed air flow, below about 300–350 mph,
the equation form of the conservation of energy principle is also known as Bernoulli’s law. Readers may
have seen this equation written with a third term on
the right side of the equation that contains the height
of the fluid. In discussions treating liquid flow such as
pressure on dams or in water towers, this third term
can be very important. But in gas flow it is always
negligibly small. So, we have Bernoulli’s law for incompressible (low-speed) gas flow:
PT = constant = PS + ½ ␳V 2 = PS + q.
As an airplane flies through the atmosphere, the
atmospheric pressure is the static pressure, and total
pressure is higher than atmospheric pressure by an
amount equal to the dynamic pressure. This pressure
rise is what the car/motorcycle/boat-racing community tends to call “ram pressure.” When a model is
being tested in a wind tunnel, the fundamental situation is different. The atmospheric pressure surrounding the tunnel is generally close to the total pressure
inside the tunnel, and as the air accelerates into the
test section of the tunnel, the static pressure drops
below atmospheric pressure, again by an amount
equal to the dynamic pressure. In either case, as the
streamlines deflect to go around a body shape like a
wing, the total pressure is constant as the fluid moves
along a streamline and its static pressure changes as
THE PHYSICS TEACHER ◆ Vol. 40, March 2002
the local velocity changes. That local static pressure
becomes the surface pressure on the body shape,
which generates forces on the body. Note that local
static pressure is no longer the same as the freestream
static pressure at a modest distance away from the
body shape where the flow that has not been diverted
by the body shape.
Conservation of Momentum
Now let’s move on to conservation of momentum:
the force exerted on a fluid equals the time rate of
change (derivative with respect to time) of its linear
momentum. If you exert a force on something, you
change its momentum. If you don’t exert a force on
something, its momentum stays unchanged or is conserved. This is Newton’s laws, if you choose to call it
that. When an airfoil is producing lift, that force does
in fact change the vertical component of the airflow’s
linear momentum, and the drag force changes the horizontal component of the airflow’s linear momentum.
When the air is moving slowly enough that its density
does not change appreciably, the gist of the term in169
Fig. 2. Streamlines around the NACA 2412 airfoil.
compressible flow, then the momentum changes are
measurable as local flow velocity changes. Measuring
drag by measuring the velocity loss within the wake of
the airfoil is easily done because it is occurring over a
relatively small region of the flow field. This is a standard wind-tunnel experiment that my classes do every
semester. Measuring lift by measuring the increase in
downward vertical velocity in the flow coming off the
trailing edge of the airfoil is conceptually possible.
This downward velocity is definitely there and is
known as downwash. I have never heard of anyone
actually measuring it with sufficient precision to calculate lift, not because it is physically unsound but because it is not a practical experiment. It is not practical
because the downwash is distributed over the entire
flow field downstream of the trailing edge, and it
would thus be very difficult to measure enough data
points to integrate the distribution accurately. (A gorgeous photo of this phenomenon caused by a jet flying
However, the flow’s downward deflection angle called
downwash angle, ⑀, is routinely calculated in aerodynamics texts because it has a large effect on the flow
over the tail of the airplane. Downwash angle is easily
estimated as approximately half the angle of attack. It
is shown somewhat exaggerated in Figs. 2, 4, and 5 to
make it easier to visualize.
In the interest of generalization, it is appropriate to
recognize that the isolated wing is not the only type of
flow-field geometry. When there are other surfaces
nearby, such as walls in flow through ducts or the
ground, those other surfaces can and do change the
momentum of the flow as well. Consider the similar
cases of low pressure causing a car’s convertible top to
bulge upward or the low pressure on the roof of a
house in a hurricane that can lift the entire roof off the
house in one piece. There is definitely lift being produced in these situations, but without net downwash.
170
How can that be? Upward lift on the roof causes a
downward momentum change, but it is almost immediately canceled by the upward momentum change associated with the downward force on the ground.
This must be so because once the air has passed the
obstacle and the local flow-pattern variations settle
out, the flow cannot end up anything except parallel to
the ground. That is, the stabilized flow cannot penetrate the ground, so there cannot be any velocity component perpendicular to the ground.
Conservation of Mass
And last but not least, the conservation of mass:
the amount of mass per second flowing between
streamlines remains constant. The effect of squeezing
streamlines together as they divert around the front of
an airfoil shape is that the velocity must increase to
keep the mass flow constant since the area between the
streamlines has become smaller. After the flow has
passed the thickest part of the airfoil and the streamlines begin to spread out as they approach the trailing
edge, the velocity decreases. Exercising the conservation-of-mass principle is what you do when you put
your thumb over the open end of a garden hose to
make the water squirt out faster. Smaller area means
faster velocity. This HAS TO BE INCLUDED with
Bernoulli’s law when explaining lift for it to really
make sense.
Now we can finally get down to the detailed explanation of how lift is generated! First, let’s pause for a
moment to point out the flaw in an old description
that is still sometimes used, the “equal-time-of-passage” concept. In this concept the relation between air
speed and static pressure expressed as Bernoulli’s law is
usually stated correctly. But do you remember hearing
over the curved top surface having to go faster than the
particles that went underneath, because they have a
longer path to travel but must still get there at the
same time? This is simply not true. It does not happen. The fact is that the particles that went the long
route over the top go so much faster due to streamline
squeezing that they get to the trailing edge before the
ones that took the shortcut underneath. I have
demonstrated this in a water tunnel many times using
intermittently pulsed hydrogen bubbles to make the
flow visible. And FoilSim can easily be made to show
THE PHYSICS TEACHER ◆ Vol. 40, March 2002
the result of theoretical calculations that verify the
same thing.
Lift
The proper explanation of lift generation is a bit
more complex, but we now have the background to
appreciate it. Shown in Fig. 2 is a plot of an airfoil
with a nearly flat bottom surface, the NACA 2412.
Most single-engine Cessna aircraft have used this airfoil since the 1940s, which almost certainly makes it
the most widely used airfoil in the world. The equaltime-of-passage picture of lift-producing flow patterns
leads to the seemingly logical yet incorrect conclusion
that this airfoil could not produce upward lift when it
is inverted. I have the very similar flat-bottom NACA
4412 airfoil on my own aerobatic aircraft, and I routinely fly it upside down. I can assure the readers that
lift by an inverted airfoil does work. The question
then is what key factor has been left out of the simplified picture. The missing item is called the stagnation
point, the small region near the leading edge at which
the local velocity stagnates or is brought to a standstill
against the airfoil surface. It is literally the point at
which the flow field splits. Flow above the stagnation
point flows along the upper surface, and flow below
the stagnation point flows along the lower surface.
Critical to this picture is that for a cambered airfoil the
stagnation point is not right at the leading edge. It is a
little below the leading edge, displaced a couple percent of the chord length aft of the leading edge on the
lower surface of the airfoil. The result is that the split
flow field is interacting with an effective surface shape
that is not nearly as simple as the airfoil shape drawn at
zero angle of attack.
The precise position of the stagnation point is determined by the magnitude of the circulation. The
circulation concept, which comes up in yet a third way
of calculating lift, is that if one were to remove the uniformly distributed freestream velocity from the curving flow around a lifting airfoil, what would remain is
a rotating, roughly elliptical flow pattern that moves
clockwise around the airfoil given the direction of flow
is called circulation. Its existence is not intuitively easy
to accept, but I have seen it nicely photographed by a
camera moving exactly with the freestream flow. The
strength of the circulation is directly proportional to
THE PHYSICS TEACHER ◆ Vol. 40, March 2002
Fig. 3. Surface pressure.
Fig. 4. Symmetric airfoil at positive angle of attack.
Fig. 5. Inverted NACA 2412 airfoil.
the magnitude of the lift force, known as the KuttaJoukowski law. This fact is the basis of Prandtl’s lifting
line theory, the other lift calculation process previously
mentioned. There is also a boundary condition called
the Kutta condition, which specifies that there must be
a rear stagnation point that remains exactly at the trailing edge of the airfoil. These two facts taken together
are the physical requirement that causes the front stagnation point to move farther and farther aft along the
lower surface as the angle of attack is increased.
The streamlines are squeezed together as the flow
diverts around the nose of the airfoil, increasing the local velocity in accordance with the conservation of
mass. The velocity increase decreases the local static
pressure, which is also the surface pressure on the airfoil, in accordance with Bernoulli’s law which you recall is the conservation of energy. Then aft of the
171
thickest part of the airfoil shape the streamlines spread
back out. The flow slows down, and the local static
pressure goes back up. This happens to some extent
on both the upper and lower surface of the airfoil, but
it is much more pronounced on the forward portion of
the upper surface, so the upper surface gets the credit
for being the primary lift producer.
Once this concept is clear, then it is also clear that
the surface pressure distribution will vary along the
chord length of the airfoil, as illustrated in Fig. 3.
Note that FoilSim will display either the pressure distribution or the local velocity distribution very nicely.
This pressure distribution is also dramatically different
at different angles of attack. Yet the equal-time-of-passage discussion of how Bernoulli’s law relates to lift ignores the pressure variation along the chord. This implies to many readers that pressure is constant, another
reason the oversimplified presentation fails to explain
logically some aerodynamic details. The resultant
force is determined by integrating the surface-pressure
distribution over the surface area of the airfoil. The integration process might be more detailed than a high
school class is prepared for. But measuring and integrating the pressure distribution is another standard
experiment in my wind-tunnel testing course, which is
predominately junior-level engineering students. And
Because of the stagnation-point position, the typical cambered airfoil needs to be oriented 2 or 3⬚ nose
down, at negative angle of attack, to get it to produce
zero lift. The stagnation point is also not right at the
leading edge of a symmetric airfoil, one without camber, when the airfoil is at nonzero angle of attack and is
producing lift. This is shown in Fig. 4. This answers
the apparent mystery of how a symmetric airfoil can
produce lift. As far as the flowing air molecules know,
the airfoil is not a symmetric shape. This is also true of
a flat plate at non-zero angle of attack. On these
shapes as well, as angle of attack increases, the stagnation point moves further aft along the lower surface.
And imagine also this same phenomenon when the
airfoil is upside down. As you can see readily in Fig. 5,
the streamlines illustrate that the flow over the original
bottom surface, which is now the top surface, is definitely not encountering a flat surface when the airfoil
is set at a modest angle of attack. There is lots of curvature near the leading edge of what is now the top
172
surface, causing the inverted airfoil to produce upward
lift. The disadvantage of using the airfoil shape in this
orientation is that the shape is now poor for lift production, so the angle of attack must be significantly
higher than that required to produce the same lift
when it is right-side up. The result is that drag is high
and aerodynamic efficiency (usually expressed as the
ratio of lift to drag) thus is low. But the lift production
is definitely there.
Bernoulli and/or Newton
So, where has this modestly complex verbal trip
taken us? Here’s my mental picture. Two things are
happening simultaneously as an airfoil produces lift.
If we take the flow-field detail perspective, we use the
conservation of mass and conservation of energy,
Bernoulli’s law, to describe a streamline-squeezing pattern that produces low pressure on the airfoil surface,
resulting in an upward force on the airfoil. In the larger-scale perspective, as long as there are no other flowaltering surfaces nearby, the forces on the airfoil are
acting on the moving fluid and changing its momentum in accordance with the conservation-of-linearmomentum principle, Newton’s laws. Both pictures
can be expressed as mathematical models that correctly
calculate the forces being generated. Which one is
preferable depends only on which one is simpler to use
with the data available. Neither is inherently more accurate or more correct.
I would like to conclude with a plea to teachers to
emphasize whichever model works more conveniently
in their scenario, without stating or even implying that
the other is wrong. I always explain lift in terms of
Bernoulli’s law and have felt comfortable that it made
sense to audiences at many different levels. I carefully
reviewed several oft quoted references4-7 in the
physics-teaching literature and do not feel that any of
them describe a shortcoming of Bernoulli’s law that is
technically correct. Besides that, Bernoulli’s law is one
of the foundations of fluid physics and is the source of
some of my favorite aerodynamic-toy demonstrations.
However, I readily agree that Newton’s laws may be a
simpler description as long as one does not need to
evaluate the details of the flow field. My hat’s off to
those of you who are accepting the Herculean challenge of finding a way to describe complex physical
principles in sufficiently simple terms to hit home
THE PHYSICS TEACHER ◆ Vol. 40, March 2002
with students whose attention spans may be pretty
short and who may not be prepared to follow an explanation that uses calculus. But Bernoulli is an icon of
aerodynamics and deserves to be so. There is no need
for controversy.
References
1. George Gerhab and Charles Eastlake, “Boundary layer
control on airfoils,” Phys. Teach. 29, 150–151 (March
1991).
2. FoilSim, an eye-catching and user-friendly PC program
that calculates and displays the flow field around airfoils. Used throughout Ref. 3 and downloadable free
from its website.
3.
4.
5.
“Beginner’s Guide to Aerodynamics,” a publication of
NASA Lewis (now NASA Glen) available on the web at
http://www.grc.nasa.gov/www/K-12/airplane/bga.html.
An incredible resource for teaching materials, even to
downloadable slides for illustrating lectures. Also examines the three most common incorrect explanations of
lift.
Norman Smith, “Bernoulli and Newton in fluid mechanics,” Phys. Teach. 10, 451–455 (Nov. 1972).
Jef Raskin, “Foiled by the Coanda Effect,” Quantum 5,
THE PHYSICS TEACHER ◆ Vol. 40, March 2002
6.
7.
5–11 (Sept./Oct. 1994).
Klaus Weltner, “A comparison of explanations of the
aerodynamic lifting force,” Am. J. Phys. 55, 50–54 (Jan.
1987).
Chris Waltham, “Flight without Bernoulli,” Phys. Teach.
36, 457–462 (Nov. 1998).
elementary aerodynamics are:
•
Hubert Smith, The Illustrated Guide to Aerodynamics,
2nd ed. (Tab Books, 1992). An excellent basic book
written by a close friend of mine and longtime teacher at
Penn State.
•
John Anderson, Introduction to Flight, 2nd ed. (McGraw-Hill, 1985). More technical in presentation than
Smith’s. The author is widely recognized as the best
textbook writer in the aerodynamics field and has written several other books.
•
Ira Abbott and Albert von Doenhoff, Theory of Wing
Sections (Dover Publications, 1959). Old, but considered the bible of airfoil theory. Much of it is the same as
NACA (National Advisory Committee for Aeronautics,
the predecessor of NASA) Technical Report 824, which
is available at http://naca.larc.nasa.gov/.
173
```
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